It would appear that Tim's question has hit one of FOM's more sensitive
nerves.
I may be repeating points already made, but it seems to me that before
one can judge such properties of the question as whether it is
tautological the question should be split into two.
1. Can mathematics be formalized?
2. Is there a universal mathematical framework?
Question 1 has surely driven the form of mathematics for millennia.
Question 2 on the other hand seems to be largely a 20th century
question. One might argue that Descartes addressed it three centuries
earlier with his algebraization of geometry, but that did little for
topology and number theory.
Tim began with the hypothesis of ZFC as a positive answer to 2. However
the pro-tautologists seem to be addressing 1.
The 2+2 = 4 example has been suggested as a possible counterexample to
question 2. Its interpretation in terms of long strings of curly
braces, and its Mizar proof, merely served to knock it out as a
counterexample, so we must look elsewhere for counterexamples to 2.
It seems even less likely as a counterexample to 1; how much more formal
can one get than 2+2 = 4?
The question I want to raise in this post is, is 2+2 = 4 even mathematics?
Is a red ball a red ball? If we interpret the first instance of "red
ball" as connoting red balls and the second as connoting those entities
that are both red and a ball, there is more to this question than to "is
a ball a ball," especially when compared with the corresponding analysis
of "is a bread knife a bread knife" meaning "is a bread knife something
that is both bread and a knife?"
If we write 2 and 4 in Roman numerals, namely II and IIII (clock style)
respectively, and use concatenation to denote addition rather than
multiplication, we can ask whether IIII equals IIII, reading the first
as the sum of II and II and the second as the Roman numeral for 4.
Now there is content in this question in just the same sense as we found
content in "is a red ball a red ball?" The ball question is surely not
mathematical in any way, yet the form of the question is the same as for
the above rendition of "is 2+2 = 4?", with an elided conjunction symbol
in place of an elided summation symbol.
If we take the number IIII to mean the sum of II and II in the same way
we take "red ball" to mean the conjunction of "red" and "ball", then
what makes "is IIII equal to IIII?" mathematical and "is a red ball a
red ball?" nonmathematical resides only in how we designate their terms:
II is a mathematical term, "red" and "ball" are nonmathematical.
Is either kind of term more formal than the other? Hardly: both have
concrete significance---an Urbis Romana waiter miscomputing II+II can
forget about his tip---yet both admit formalization, witness the Boolean
algebra of conjunction.
But the reasoning by which we infer the truth of "is IIII equal to
IIII?" seems too close to that for "is a red ball a red ball?" to
justify treating one as mathematical and the other as nonmathematical on
any basis other than their respective domains of discourse. But if the
domains are significant we would then want to make all sorts of
distinctions, for example that the latter question deals with inanimate
objects such as balls while "is a tall man a tall man?" deals with
animate objects.
I submit that the essential content of 2+2 = 4 is not mathematical but
logical. The fact that numbers are considered mathematical while
predicates are not (by mathematicians if not logicians) has little
bearing on the essential content of 2+2 = 4, which is no more than the
easily verified IIII = IIII suitably understood.
(Contrary to the widespread impression that Russell and Whitehead prove
1+1 = 2 on page 362 of Volume I, they only promise it there and do not
actually prove it until the top of page 83 (2nd. Ed.) of Volume II,
where they promise three applications for it. The myriad dense formulas
of Volumes I and II leading up to that result arguably demonstrate a
respect for mathematics best characterized as unhealthy.)
Vaughan Pratt